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That there could be three chosen faces of one color and the rest all different colors implies there are at least five solids. That the chosen faces could all be different colors then implies there are more than four faces on each solid. Therefore, there are 6, 8, 12 or 20 faces on each solid. There also cannot be more solids than the number of faces on each one, as there is a definite probability that all the colors are different.
If F is the number of faces on each solid and N is the number of solids, then the probability the chosen colors are all different is F!/((F-N)!*F^N). The probability that three are the same and the rest all different is C(N,3)*(F-1)!/((F-N+2)!*F^(N-1)). The only case where the ratio of these is 2/3 is where F = 20 and N = 11:
20!/((20-11)!*20^11) = 26189163/800000000
C(11,3)*(20-1)!/((20-11+2)!*20^(11-1) = 165*19!/(11!*20^10)
= 78567489/1600000000
The ratio of these is 2/3.
There are 11 20-sided solids (icosahedra).
Based on Enigma No. 1662, "Red face", by Susan Denham, New Scientist, 3 September 2011, page 32. |
